Question: $ -1.\overline{1} \div 0.\overline{5} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -11.1112...\\ x &= -1.1112...\end{align*} $ $\begin{align*} 9x &= -10 \\ x &= -\dfrac{10}{9}\end{align*} $ $\begin{align*} 10y &= 5.5555...\\ y &= 0.5555...\end{align*} $ $\begin{align*} 9y &= 5 \\ y &= \dfrac{5}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{10}{9} \div \dfrac{5}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{10}{9} \times \dfrac{9}{5} = {?} $ $ \phantom{-\dfrac{10}{9} \times \dfrac{5}{9}} = \dfrac{-10 \times 9}{9 \times 5} $ $ \phantom{-\dfrac{10}{9} \times \dfrac{5}{9}} = \dfrac{-10 \times \cancel{9}} {\cancel{9} \times 5} $ $ \phantom{-\dfrac{10}{9} \times \dfrac{5}{9}} = -\dfrac{10}{5} $ Simplify: ${= -2}$